More general families of infinite graphs

More generl fmilies of infinite grphs Antoine Meyer Forml Methods Updte 2006 IIT Guwhti

Prefix-recognizle grphs Theorem Let G e grph, the following sttements re equivlent: G is defined y reltions of the form (U V ) Id W G is the prefix rewriting grph of recognizle rewriting system (+ regulr restriction) G is the trnsition grph of pushdown utomton with ε-trnsitions G is the result of unfolding finite grph nd pplying regulr sustitution

Prefix-recognizle grphs (2) Properties Rechility reltions nd sets of rechle vertices re effectively computle The lnguges of prefix recognizle grphs re the context-free lnguges The mondic second order theory of ny prefix-recognizle grph is decidle

Extensions nd vrints More generl rewriting systems (term rewriting systems) More generl computtion models (higher-order pushdown utomt) More powerful or iterted trnsformtions More generl finitely presented inry reltions (utomtic or rtionl reltions)

Term rewriting Term: expression over function symols nd vriles f f (g(x),f (k,y)) g f ((c(x)))) c x k y x Term rewriting system: set R of pirs of terms (l,r) Ground rewriting: l r for ll (l,r) R

Ground term rewriting grphs Definition A grph G is (recognizle) ground rewriting grph if ech of its set of edges is the ground rewriting reltion of (recognizle) ground term rewriting system R. Exmple The two-dimensionl grid Properties Rechility reltions nd sets of rechle vertices re effectively computle The first order theory with rechility of ny ground rewriting grph is decidle The lnguges of ground rewriting grphs re...?

Definition Higher-order stck: Higher-order pushdown stcks level 1: sequence of stck symols (ordinry stck) level n: (non-empty) sequence of level n 1 stcks Exmple c Level 1 store s with on top c c c Level 2 store with s on top

Higher-order pushdown utomt Definition Higher-order (level n) pushdown utomton: pushdown utomton + higher order opertions push k : duplicte top-most level k stck pop k : destroy top-most level k stck Exmple Automton ccepting the lnguge {ww w N } Interesting strct model for higher-order recursive sequentil progrms (e.g. ML, Scheme,...)

Higher-order prefix-recognizle grphs Theorem Let G e grph, the following sttements re equivlent: G is the trnsition grph of level n pushdown utomton G is the result of pplying (unfolding + sustitution) n times to finite grph Properties Rechility reltions nd sets of rechle vertices re effectively computle The lnguges of these grphs re the IO lnguges The mondic second order theory of ny such grph is decidle

A grph c c c c

Rtionl reltions Definition A inry reltion over words is clled rtionl if it is the set of pirs ccepted y finite trnsducer Exmple A/A B/B ε/a q 0 q 1 ccepts the reltion {(A n B m,a n+1 B m ) m,n 0}

Rtionl grphs Definition A rtionl grph is grph whose edge reltions re rtionl Properties Rechility reltions nd sets of rechle vertices re non-recursive The lnguges of rtionl grphs re the context-sensitive lnguges There exist some rtionl grphs whose first order theory is undecidle

A hierrchy of infinite utomt regulr finite context-free pushdown, prefix-recognizle... context-sensitive rtionl recursively enumerle computle

Appendix: lnguges of rtionl grphs (proof)

Sufmilies of rtionl grphs Synchronized trnsducer: ll runs of the form 1 / 1 q 0... n/ n ε/ n+1... ε/ n+k qf 1 / 1 or q 0... n/ n n+1/ε... n+k /ε q f Automtic grphs: rtionl grphs defined y synchronized trnsducers Synchronous trnsducer: no ε ppering on ny trnsition Synchronous grphs: rtionl grphs defined y letter-to-letter trnsducers

Lnguges of rtionl grphs Existing proof uses the Penttonen norml form for context-sensitive grmmrs Techniclly non-trivil No link to complexity No notion of determinism Our contriutions: New syntcticl proof using tiling systems Chrcteriztion of lnguges for su-fmilies of grphs Chrcteriztion of grphs for su-fmilies of lnguges

Tiling systems Definition A frmed tiling system is finite set of 2 2 pictures (tiles) with order symol # Picture: rectngulr rry of symols Picture lnguge of : set of ll frmed pictures with only tiles in Word lnguge of : set of ll first row contents in the picture lnguge of Proposition [Ltteux&Simplot97] The lnguges of tiling systems re precisely the context-sensitive lnguges

A tiling system # # # # # # # # # # # # # # # # # # # # # # # #

Proof technique Proof in three steps: 1 Trce-equivlence of rtionl nd synchronous grphs 2 Simultion of synchronous grph y tiling systems 3 Simultion of tiling system y synchronous grph 1 Rtionl Synchronous Tiling system 2 1 relies on elimintion of ε in trnsducers 2 nd 3 rely on identifying sequences of vertices with pictures 3

Rtionl synchronous grph Proof ide Allow ll trnsducers sttes to idle (ε/ε loops) Mterilize ε s fresh symol # ( synchronous grph) Define I nd F s the shuffle of i nd F y #

Rtionl synchronous grph B/B B/B ε A A 2 T : q 0 ε/a q 1 B AB A 2 B T : A/A q 0 ε/a B/B q 1 B 2 AB 2 A 2 B 2

Rtionl synchronous grph ε/ε A/A ε/ε B/B T : q 0 ε/a q 1 ε/ε A/A ε/ε B/B T : q 0 ε/a q 1

Rtionl synchronous grph #/# A/A #/# B/B T : q 0 #/A q 1 #/# A/A #/# B/B T : q 0 #/B q 1

Rtionl synchronous grph T : #/# A/A q 0 #/# A/A #/A #/# B/B q 1 #/# B/B ### ##B #B# B## ##A #A# A## #AB A#B AB# #AA A#A AA# AAB AAA T : q 0 #/B q 1 #BB B#B BB# ABB BBB

Synchronous grph tiling system Proof ide Identify grph vertices nd picture columns Estlish ijection etween ccepting pths nd pictures Deduce ijection etween synchronous grphs nd tiling systems 1 n v 1 0 2 v1 n vn v 0 v 1 v n